Recent and promising advancements in fields ranging from materials science to quantum physics are now being used to produce new quantum-system-based technologies. In particular, certain quantum systems can be used to encode and transmit quantum information. Quantum systems comprising just two discrete states, represented by “|0” and “|1” can potentially be employed in a variety of quantum-system-based applications including quantum information encoding and processing, optical quantum lithography, and metrology. In general, a quantum system comprising two discrete states is called a “qubit system,” and the states |0 and |1 called “qubit basis states,” can also be represented in set notation as {|0|1. Vertically and horizontally polarized photons are examples of basis states of a two-state quantum system based on electromagnetic radiation. A qubit system can exist in the state |0 the state |1 or in any of an infinite number of states that simultaneously comprise both |0 and |1 which can be mathematically represented as a linear superposition of states:|ψ=α|0+β|1The state |ψ is called a “qubit,” and the parameters α and β are complex-valued coefficients satisfying the condition:|α|2+|β|2=1
Performing a measurement on a quantum system is mathematically equivalent to projecting the state of the quantum system onto one of the basis states, and, in general, the probability of projecting the state of the quantum system onto a basis state is equal to the square of the coefficient associated with the basis state. For example, when the state |ψ of the qubit system is measured in the basis {|0 |1, one has a probability |α|2 of finding the quantum system in the state |0 and a probability |β|2 of finding the quantum system in the state |1
The infinite number of pure states associated with a qubit system can be geometrically represented by a unit-radius, three-dimensional sphere called a “Bloch sphere”:
            ❘        ⁢    ψ    ⁢          〉        =                    cos        ⁡                  (                      θ            2                    )                    ⁢              ❘            ⁢      0      ⁢              〉              +                  ⅇ        ⅈϕ            ⁢              sin        ⁡                  (                      θ            2                    )                    ⁢              ❘            ⁢      1      ⁢              〉            where0≦θ<π, and0≦φ<2π.FIG. 1 illustrate a Bloch sphere representation of a qubit system. In FIG. 1, lines 101-103 are orthogonal x, y, and z Cartesian coordinate axes, respectively, and a Bloch sphere 106 is centered at the origin. There are an infinite number of points on the Bloch sphere 106, each point representing a unique state of a qubit system. For example, a point 108 on the Bloch sphere 106 represents a unique state of a qubit system that simultaneously comprises, in part, the state |0 and, in part, the state |1 However, once the state of the qubit system is measured in the basis {|0|1, the state of the qubit system is projected onto the state |0 110 or onto the state |1 112.
The state of a system comprising two or more qubit systems is represented by a tensor product of qubits, each qubit associated with one of the qubit systems. For example, consider a system comprising a first qubit system and a second qubit system that is represented by the state:|ψ12=|ψ1|ψ2 where the state of the first qubit system is:
            ❘        ⁢    ψ    ⁢                  〉            1        =                    1                  2                    ⁢              (            ⁢              ❘            ⁢      0      ⁢                        〉                1              +                  ❘            ⁢      1      ⁢                        〉                1            ⁢              )            and the state of the second qubit system is:
            ❘        ⁢    ψ    ⁢                  〉            2        =                    1                  2                    ⁢              (            ⁢              ❘            ⁢      0      ⁢                        〉                2              +                  ❘            ⁢      1      ⁢                        〉                2            ⁢              )            The state |ψ2 can also be rewritten as a linear superposition of states:
                                          ❘                    ⁢          ψ          ⁢                                    〉                        12                          =                ⁢                              ❘                    ⁢          ψ          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          ψ          ⁢                                    〉                        2                                                  =                ⁢                                            1              2                        ⁢                          (                        ⁢                          ❘                        ⁢            0            ⁢                                          〉                            1                        ⁢                          ❘                        ⁢            0            ⁢                                          〉                            2                                +                                    ❘                        ⁢            0            ⁢                                          〉                            1                        ⁢                          ❘                        ⁢            1            ⁢                                          〉                            2                                +                                    ❘                        ⁢            1            ⁢                                          〉                            1                        ⁢                          ❘                        ⁢            0            ⁢                                          〉                            2                                +                                    ❘                        ⁢            1            ⁢                                          〉                            1                        ⁢                          ❘                        ⁢            1            ⁢                                          〉                            2                        ⁢                          )                                          where the terms |01|02, |01|12, |11|02, and |11|12 are basis of the tensor product space. Each product state in the state |ψ12 has an associated coefficient of ½, which indicates that when the state of the first qubit system is measured in the bases {|01,|11}, and the state of the second qubit system is measured in the basis {|02,|12}, there is a ¼ (|½|2) probability of the combined qubit systems being found in any one of the product states. Certain state of the combined qubit systems, however, cannot be represented by a product of associated qubits. These qubit systems are said to be “entangled.” Quantum entanglement is a property of quantum mechanics in which the states of two or more quantum systems are correlated, even though the quantum systems can be spatially separated. An example entangled state representation of an entangled two-qubit system is:
            ❘        ⁢          ψ      +        ⁢                  〉            12        =                    1                  2                    ⁢              (            ⁢              ❘            ⁢      0      ⁢                        〉                1            ⁢              ❘            ⁢      1      ⁢                        〉                2              +                  ❘            ⁢      1      ⁢                        〉                1            ⁢              ❘            ⁢      0      ⁢                        〉                2            ⁢              )            The entangled state |ψ+12 cannot be factored into a product of the qubits α1|01+β1|11 and α2|02+β2|12, for any choice of the parameters α1, β1, α2, and β2.
The state of an un-entangled, two-qubit system can be distinguished from the state of an entangled, two-qubit system as follows. Consider an un-entangled, two-qubit system in the state |ψ12. Suppose a measurement performed on the first qubit system in the basis {|01,|11} projects the state of the first qubit system onto the state |01. According to the state |ψ12, the state of the un-entangled, two-qubit system immediately after the measurement is the linear superposition of states (|01|02+|01|12)/√{square root over (2)}. When a second measurement is performed on the second qubit system in the basis {|02,|12} immediately following the first measurement in an identical reference frame, there is a ½ probability of projecting the state of the second qubit system onto the state |02 and a ½ probability of projecting the state of the second qubit system onto the state |12. In other words, the state of the second qubit system is not correlated with the state of the first qubit system.
In contrast, consider an entangled, two-qubit system in the entangled state |ψ+12. Suppose that a first measurement performed on the first qubit system in the basis {|01,|11} also projects the state of the first qubit system onto the state |01. According to the entangled state |ψ+12, the state of the entangled, two-qubit system after the first measurement is the product state |01|12. When a second measurement is performed on the second qubit system in the basis {|02,|12}, the state of the second qubit system is |12 with certainty. In other words, the state of the first qubit system |01 is correlated with the state of the second qubit system |12.
Entangled quantum systems have a number of different and practical applications in fields ranging from quantum computing to quantum information processing. In particular, polarization entangled-photons can be used in quantum information processing, quantum cryptography, teleportation, and linear optics quantum computing. The term “photon” refers to a single quantum of excitation of a mode of the electromagnetic field. Note that an electromagnetic wave comprises both an electric field component, {right arrow over (E)}, and an orthogonal magnetic field component, {right arrow over (B)}. However, because the amplitude of the magnetic field component B0 is smaller than the amplitude of the electric field component E0 by a factor of 1/c, where c represents the speed of light in free space (c=3.0×108 m/sec), the electric field component accounts for most the electromagnetic wave interactions with matter. As a result, a polarization state of an electromagnetic wave is typically represented by the electric field component alone.
FIGS. 2A-2B illustrates vertically and horizontally polarized photon basis states, respectively, that can be used as basis states for polarization entangled-photons. In FIGS. 2A-2B, vertically and horizontally polarized photons are represented by oscillating continuous waves propagating along z-coordinate axes 202 and 204, respectively. As shown in FIG. 2A, a vertically polarized photon |V oscillates in the yz-plane. Directional arrow 206 in xy-plane 208 represents one complete oscillatory cycle as |V advances along the z-coordinate axis 202 through one complete wavelength. In FIG. 2B, a horizontally polarized photon |H oscillates in the xz-plane. Directional arrow 210 in xy-plane 212 represents one complete oscillatory cycle as |H advances along the z-coordinate axis 204 through one complete wavelength.
The Bell states:
                              ❘                ⁢                  ψ          -                ⁢                  〉                    =                                    1                          2                                ⁢                      (                    ⁢                      ❘                    ⁢          H          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          V          ⁢                                    〉                        2                          -                              ❘                    ⁢          V          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          H          ⁢                                    〉                        2                    ⁢                      )                                ,                  ⁢                            ❘                ⁢                  ψ          +                ⁢                  〉                    =                                    1                          2                                ⁢                      (                    ⁢                      ❘                    ⁢          H          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          V          ⁢                                    〉                        2                          +                              ❘                    ⁢          V          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          H          ⁢                                    〉                        2                    ⁢                      )                                ,                  ⁢                            ❘                ⁢                  ϕ          -                ⁢                  〉                    =                                    1                          2                                ⁢                      (                    ⁢                      ❘                    ⁢          H          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          H          ⁢                                    〉                        2                          -                              ❘                    ⁢          V          ⁢                                    〉                        1                    ⁢                      ❘                    ⁢          V          ⁢                                    〉                        2                    ⁢                      )                                ,    and                      ❘            ⁢              ϕ        +            ⁢              〉              =                            1                      2                          ⁢                  (                ⁢                  ❘                ⁢        H        ⁢                              〉                    1                ⁢                  ❘                ⁢        H        ⁢                              〉                    2                    +                        ❘                ⁢        V        ⁢                              〉                    1                ⁢                  ❘                ⁢        V        ⁢                              〉                    2                ⁢                  )                    where
subscript “1” represents a first transmission channel; and
subscript “2” represents a second transmission channel.
are examples of polarization entangled-photons that can be used in a number of different entangled-state applications.
Although polarization-entangled photons have a number of potentially useful applications, polarization-entangled photon sources typically cannot be practically implemented in a wide variety of entangled state applications. For example, in “New High-Intensity Source of Polarization-Entangled Photon Pairs,” by Kwiat et al., Physical Review Letters, vol. 75, 4337, (1995), Kwiat describes a high-intensity source of polarization entangled-photon Bell states that works for continuous electromagnetic waves but not for electromagnetic wave pulses. In addition, only photons emitted in a particular direction are entangled. As a result, only a limited number of Bell states can be generated. In “Ultrabright source of polarization-entangled photons,” by Kwiat et al., Physical Review A, vol. 60, R773, (1999), Kwiat also describes a source of polarization-entangle photon pairs. However, thin crystals and continuous wave pumps have to be used in order to obtain good entanglement. In “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” by Taehyun Kim et al., Physical Review A, vol. 73, 012316 (2006) and in “Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints,” by Fiorentino et al., Physical Review A, vol. 69, 041801(R) (2004), both Kim and Fiorentino describe an ultrabright parametric down-conversion source of Bell state polarization-entangled photons. However, these polarization-entangled photon sources cannot be used in microscale applications, are expensive to produce, and need periodic adjustments. Physicist, computer scientist, and entangled state users have recognized a need for polarization entangled-photon sources that are compatible with both continuous wave and pulse pump sources and can be coupled to fiber optic couplers for implementation in microscale devices.